supplement to lab activity #13: a simple stella model of

ERTH 535, Dr. Dave Dempsey “Planetary Climate Change” Dept. of Earth & Climate Sciences

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Supplement to Lab Activity #13: A Simple STELLA Model of the Planetary Heat Budget

with a Greenhouse Effect I. Equations for a Heat Budget Model of a Planet with a Fully

Transparent Atmosphere

In the simplest version of our STELLA model, the atmosphere is completely transparent to all wavelengths of radiation and does not exchange heat in any way with the surface. As a result, the atmosphere plays no role in the model, and we represent a heat budget only for a layer at the surface of the planet, globally averaged. (There is no greenhouse effect.)

This model consists of eleven coupled equations: ** Principle of Conservation of Energy (expressed in terms of a heat budget equation for the surface layer):

(1)

where: rate at which the heat content of the surface layer changes with respect to time ( ). (Here, a small increment of time, and

the change in heat content of the layer over that period of time. The notation “ ” emphasizes that the heat content varies with time.) rate at which the surface absorbs solar radiation rate at which the surface emits longwave infrared (LWIR) radiation ** A relation between solar radiation arriving at and absorbed by the planet:

(2) where: albedo of the surface

rate at which solar radiation arrives at the top of the atmosphere (and also the surface, since the model atmosphere is transparent)

ΔHSfc t( )Δt

= SRabsSfc − ERSfc

ΔHsfc t( ) Δt ≡t Δt ≡

ΔHsfc t( ) ≡Hsfc t( )

SRabsSfc ≡ERSfc ≡

SRabsSfc = 1−α Sfc( )× SRarrα Sfc ≡

SRarr ≡


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** The relation between rate and flux, applied to arriving solar radiation: (3)

where: the cross-sectional area of the planet, the effective area that the planet presents to the sun’s rays the flux of solar radiation directly facing the sun at distance from the sun ** The Inverse Square Law, applied to solar radiation:

(4) where: the solar constant, the solar radiative flux directly facing the sun at the planet’s average distance from the sun

the distance between the planet and the sun ** Relation between rate and flux, applied to radiative emission:

(5)

where: surface area of a spherical planet emission flux of LWIR radiation from the surface ** The Stefan-Boltzmann Law:

(6) where: the Stefan-Boltzmann constant the time-dependent absolute temperature of the surface layer

SRarr = AX-sect × SF re2s( )

AX−sect ≡

SF re2s( ) ≡≡ re2s

SF re2s( ) = SF re2s( )× re2s re2s( )2


re2s ≡

ERSfc = ASfc × EFSfc

Asfc ≡EFSfc ≡

EFSfc =σ TSfc (t)⎡⎣ ⎤⎦4

σ ≡TSfc (t) ≡


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** Relation between change in temperature and change in heat content of the surface layer. This is a proportionality in which the constant of proportionality is , where heat capacity = :

From this we can develop a version that relates absolute temperature of

the surface layer to the total heat content of the layer:

(7) ** Relation between mass, volume ( ), and density ( ) of the layer:

(8)

** Relation between volume, depth ( ), and area of the surface layer:

(9) ** Relation between the radius ( ) and surface area of the spherical planet:

(10) ** Relation between the radius and cross-sectional area of the spherical

planet (which is just the relation between radius and area of a circle):

(11)

These 11 equations collectively constitute our simple model. To solve the equations using STELLA modeling software, the values of

the following physical quantities (which we can call “parameters”), which appear in the model equations above, must be specified: , , , , , , , and These parameters can, in principle, vary from one physical context to another.

The value of the Stefan-Boltzmann constant, , must also be specified.

1 CH CH ≡ mSfc × cH

ΔTSfc = ΔHSfc mSfc × cH( )

TSfc = HSfc mSfc × cH( )VSfc ρSfc

mSfc = ρSfc ×VSfc

dSfc

VSfc = dSfc × ASfc

Re

ASfc = 4πRe2

AX−sect = πRe2

Re dSfc ρSfc cH α Sfc re2s re2s SF re2s( )

σ


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Finally, we also must specify an “initial”, or starting value of the heat content of the surface layer, (that is, the heat content of the surface layer at time t = 0.) All other quantities can be calculated from these nine physical parameters plus one constant plus one initial value. II. Equations for a Model with Separate Heat Budgets for the Surface

Layer and the Atmosphere A more sophisticated version of the planetary heat budget model adds an

atmosphere (treated as a single layer with a single temperature), which can absorb and reflect solar radiation and absorb and emit terrestrial (longwave infrared) radiation. Among other things, the surface layer and the atmosphere influence each other by absorbing radiation emitted by the other, which creates the potential for simulating (crudely) a greenhouse effect. (However, like the simpler, surface-layer only model, we ignore transfer of heat between the surface and atmosphere via conduction and evaporation & condensation and continue to focus only on radiative transfers of energy.)

The equations for this version of the model are:

** Principle of Conservation of Energy (written as heat budget equations):

(1)(a) For the atmosphere:

where: the rate at which the atmosphere’s heat content

changes with respect to time rate at which the atmosphere absorbs solar radiation

rate at which the atmosphere absorbs LWIR radiation emitted upward by the surface

rate at which the atmosphere emits LWIR radiation downward to the surface

rate at which the atmosphere emits LWIR radiation to space

HSfc 0( )

ΔHAtm t( )Δt

= SRabsAtm +TRabsAtm − ERAtm2Sfc − ERAtm2Space

ΔHAtm t( ) Δt ≡

SRabsAtm ≡TRabsAtm ≡

ERAtm2Sfc ≡

ERAtm2Space ≡


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(1)(b) For the surface layer:

where: rate at which the surface absorbs solar radiation

rate at which the surface absorbs terrestrial (LWIR) radiation emitted downward by the atmosphere

rate at which the surface emits LWIR radiation that is absorbed by the atmosphere

rate at which the surface emits LWIR radiation to space

Note that , where the total rate at which the surface emits LWIR radiation. For budgetary convenience, we have separated the total emission rate into a part that escapes directly to space and a part that the atmosphere absorbs.

We can modify Eqs. (1)(a) and (b) by recognizing that (that is, the surface absorbs all of the LWIR radiation emitted downward by the atmosphere, because the surface is a blackbody with respect to LWIR radiation). Moreover, (that is, the atmosphere absorbs part of the LWIR radiation that the surface emits). Making these substitutions into Eqs. (1)(a) and (b) gives modified versions:

(1)(a)* For the atmosphere:

(1)(b)* For the surface layer:

ΔHSfc t( )Δt

= SRabsSfc +TRabsSfc − ERSfc2Atm − ERSfc2Space

SRabsSfc ≡TRabsSfc ≡

ERSfc2Atm ≡

ERSfc2Space ≡

ERSfc = ERSfc2Atm + ERSfc2Space ERSfc ≡

ERabsSfc = ERAtm2Sfc

TRabsAtm = ERSfc2Atm

ΔHAtm t( )Δt

= SRabsAtm + ERSfc2Atm − ERAtm2Sfc − ERAtm2Space

ΔHSfc t( )Δt

= SRabsSfc + ERAtm2Sfc − ERSfc2Atm − ERSfc2Space


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** A relation between solar radiation arriving at and absorbed by the planet:


where: solar absorptivity of the atmosphere (that is, the fraction of solar radiation reaching the atmosphere that the atmosphere absorbs) rate at which solar radiation arrives at the top of the atmosphere


where: albedo of the surface albedo of the atmosphere

Eq. (2)(b) starts with the rate at which solar radiation arrives at the top of the atmosphere ( ). The factor then accounts for reflection and absorption in the atmosphere, which reduce the rate at which solar radiation reaches the surface. Finally, the factor accounts for reflection by the surface. What’s left is absorbed by the surface. ** The relation between rate and flux, applied to arriving solar radiation:

(3) where: the cross-sectional area of the planet, the effective area that the planet presents to the sun’s rays the flux of solar radiation directly facing the sun at distance from the sun

SRabsAtm = aAtmS × SRarr

aAtmS ≡

SRarr ≡

SRabsSfc = 1−α Sfc( )× 1−α Atm − aAtmS( )× SRarr

α Sfc ≡

α Atm ≡

SRarr 1−α Atm − aAtmS( )

1−α Sfc( )

SRarr = AX-sect × SF re2s( )

AX−sect ≡



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** The Inverse Square Law, applied to solar radiation:

(4) where: the solar constant, the solar radiative flux directly facing the sun at the planet’s average distance from the sun

the distance between the planet and the sun ** Relation between rate and flux, applied to radiative emission:


where: surface area of a spherical planet emission flux of LWIR radiation from the atmosphere

In (5)(a) we’ve taken advantage of the fact that the atmosphere is treated as a single layer with a single temperature, so the emission flux upward from the top and downward from the bottom are the same, and the surface area of the top and bottom are virtually the same (because the atmosphere is very shallow compared to the radius of the earth).


and

where: emission flux of LWIR radiation from the surface

terrestrial (LWIR) absorptivity of the atmosphere (that is, the fraction of LWIR radiation entering the atmosphere that the atmosphere absorbs)

SF re2s( ) = SF re2s( )× re2s re2s( )2


re2s ≡

ERAtm2Sfc = ERAtm2Space = ASfc × EFAtm

ASfc ≡EFAtm ≡

ERSfc2Atm = aAtmT × ERSfc = aAtm

T × ASfc × EFSfc( )

ERSfc2Space = 1− aAtmT( )× ERSfc = 1− aAtmT( )× ASfc × EFSfc( )

EFSfc ≡aAtmT ≡


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In (5)(b), represents the portion of the terrestrial (LWIR) radiation emitted by the surface that the atmosphere absorbs, while

is the rest, which is the portion of LWIR radiation emitted by the surface that the atmosphere does not absorb and hence escapes directly to space. (Note that these two rates sum to , as we’d expect.)

** The Stefan-Boltzmann Law:


where: Stefan-Boltzmann constant

terrestrial (LWIR) emissivity of the atmosphere (that is, the flux of LWIR radiation emitted by the atmosphere expressed as a fraction of the LWIR radiation that a blackbody at the same temperature would emit) (The emissivity accounts for the fact that the atmosphere is a selective absorber of LWIR radiation, and hence also a selective emitter—that is, it doesn’t emit as much as a blackbody would at the same temperature.)

is the (global avg., time-dependent) atmospheric temperature In (6)(a) we have invoked Kirchoff’s Law, which says that =

(that is, if an object absorbs a particular wavelength or wavelengths of radiation well, it is capable of emitting it (them) as well, depending on the object’s temperature).


where: the time-dependent absolute temperature of the surface

(Note that, unlike the atmosphere, the surface is effectively a blackbody for LWIR radiation, so its terrestrial (LWIR) emissivity = 1.0.)

aAtmT × ERSfc

1− aAtmT( )× ERSfc

ERSfc

EFAtm = εAtmT ×σ TAtm (t)[ ]4 = aAtmT σ TAtm (t)[ ]4

σ ≡εAtmT ≡

TAtm (t)

εAtmT aAtm

T

EFSfc =σ TSfc (t)⎡⎣ ⎤⎦4

TSfc t( ) ≡


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** Relation between absolute temperature and heat content:


where: mass of the atmosphere affected by both solar radiative

absorption and LWIR radiative emission (namely, all of it).

specific heat of air at constant pressure.


(This is the same as in the simpler version of the model.)

** Relation between mass and other quantities:


where is the total mass of the atmosphere, is the global mean sea-level atmospheric pressure, and is the acceleration of gravity.

This relation comes from the near balance between the upward force exerted by air pressure at sea level and the downward force of gravity acting on the mass of the atmosphere. (Pressure is the collective force of collisions exerted by molecules in random motion, per unit of surface area against which they collide. Weight is the force of gravity, which equals the mass of the object times the acceleration of gravity. Multiplying the global mean sea-level pressure by the surface area of the earth gives us the total upward force on air due to air pressure at sea level, while the total mass of the atmosphere times the acceleration of gravity is the total weight of air. Equating these and solving for the mass of the atmosphere gives us Eq. (8)(a).

TAtm = HAtm mAtm ×Cp( )mAtm ≡

Cp ≡

ΔTSfc = ΔHSfc mSfc × cH( )

mAtm = ASfc psl g

mAtm

pslg


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The remaining equations are the same as they were in the simpler version of the model (that is, the one without an atmosphere):


** Relation between volume, depth ( ), and area of the surface layer: (9) ** Relation between the radius ( ) and surface area of the spherical planet: (10) ** Relation between the radius and cross-sectional area of the spherical

planet (which is just the relation between radius and area of a circle):

(11)

In the simpler version of the planetary heat budget model, which effectively lacks any atmosphere at all, all of the LWIR radiation emitted to space comes from the surface, which acts as a blackbody for LWIR radiation, so the surface temperature is by definition the effective radiating temperature of the planet. However, in the version of the model with an atmosphere, LWIR radiation emitted to space now comes from both the atmosphere and the surface, which are generally not at the same temperature. So, what is the effective radiating temperature of the planet in that case?

The total radiative emission flux from the planet, , is the sum of the LWIR radiative emission fluxes from the surface directly to space and from the atmosphere to space. These two fluxes are just the respective emission rates (see Eqs. (1)(a) and (b)) divided by the surface area of the earth ( ):

mSfc = ρSfc ×VSfc

dSfc

VSfc = dSfc × ASfc

Re

ASfc = 4πRe2

AX−sect = πRe2

EFPlanet2Space

ASfc

EFPlanet2Space = ERAtm2Space + ERSfc2Space( ) ASfc


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The total emission flux from the planet ( ) can be related to the effective radiating temperature of the planet ( ) by applying the Stefan-Boltzmann Law as follows:

Expressing in terms of its constituent parts and solving for gives:

** Effective radiating temperature of the planet:

(12)

This equation is a diagnostic equation telling us about the overall physical state of the planet system, but it has no role to play in the budget otherwise (that is, no part of the planet’s heat budget depends on ).

Collectively, equations (1) through (11) above constitute our planetary

heat budget model for the planet with an atmosphere. To solve the equations (plus the diagnostic Eq. (12)) using STELLA modeling software, the values of the following 13 physical parameters must be specified: , , , , , , , , plus

, , , , and (The first eight are the same as in the surface-only version of the model.) These 13 parameters can, in principle, all vary from one context to another.

As before, the value of the Stefan-Boltzmann constant, , must also be specified, and we must also specify an “initial” (starting) value of the heat content of both the surface layer, and of the atmosphere, (that is, the heat content of the surface layer and atmosphere at time t = 0).

All other quantities can be calculated from these 13 physical parameters, one constant, and two initial values.

EFPlanet2SpaceTeffPlanet

EFPlanet2Space =σTeffPlanet4

EFPlanet2SpaceTeffPlanet

TeffPlanet = ERAtm2Space + ERSfc2Space( ) σASfc( )4

TeffPlanet

Re dSfc ρSfc cH α Sfc re2s re2s SF re2s( )psl g aAtm

T α Atm aAtmS

σ

HSfc 0( ) HAtm 0( )


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The values of seven of the thirteen physical parameters for our current earth, along with the Stefan-Boltzmann constant, are well established (though there’s no reason why you couldn’t run the model for some other planet with quite different values):

= 1004 (J/kg)/K = 9.806 m/s2 = 149.6×109 m 147.3×106 km < < 152.1×106 km

= 1368 W/m2

Several other parameters depend on the nature of the earth’s surface. For example, if the surface comprises nothing but ice-free ocean, then:

= 103 kg/m3 = 4218 J/kg = ~0.07 to 0.01 However, if the surface comprises nothing but a representative type of

rock, then might be 2.5 times higher than for water, half as great as for water, and something like 0.2 or 0.3 (or even 0.4 for desert sand). Vegetated land might have anywhere from 0.08 (coniferous forest) to 0.25 (grassland).

For a planet covered by ice, would be 0.9 to 1.0×103 kg/m3, might

be around 2000 (J/kg)/K, and might be anywhere from 0.4 (old or melting snow) to 0.8 or 0.9 (fresh snow).

The depth of the surface layer, , affected by solar heating and radiative cooling depends on the transparency of the layer and it’s viscosity of fluidity, which determines its ability to mix vertically. For a land surface, the affected depth would be relatively shallow compared to the ocean, and would depend precisely on how is defined. You might try 1 meter for a solid planet and tens of meters for a water planet (though don’t take these values as especially defensible).

Cp gre2s re2sSF re2s( ) σ = 5.67×10−8 Watts m2( ) K4

Re = 6.37×106m psl =1013.25×102 Pascals i.e., kg*m s2( ) m2⎡⎣ ⎤⎦

ρSfc cH α Sfc

ρSfc cHα Sfc

α Sfc

ρSfc cHα Sfc

dSfc

dSfc


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The terrestrial (LWIR) absorptivity of the atmosphere, , the albedo of the atmosphere, , and the solar absorptivity of the atmosphere, , can all vary from one context to another, but you can estimate them using the long-term, global average energy budget for earth that we studied in Lab Activity #5: Long-Term Average Energy Budgets for the Earth's Atmosphere and Surface. (There is information in that diagram that can help you estimate the effects of the atmosphere with and without clouds, too.)

aAtmT

α Atm aAtmS

supplement to lab activity #13: a simple stella model of

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